Idempotent Symmetry and Monogenic Functions in a Commutative Bicomplex-Type Algebra

Let A={p+Jq:p,q∈C,J2=−1} be the commutative bicomplex-type algebra in which J commutes with the scalar imaginary unit. A Cauchy–Riemann-type operator D¯ is studied on domains in C2. In the active coordinates ξ=z1−iz2 and η=z1+iz2, the equation D¯f=0 is diagonal in the idempotent basis: the e+-component is holomorphic in ξ with η as the parameter, while the e−-component is holomorphic in η with ξ as the parameter. The expression e+F(ξ)+e−G(η) is the parameter-independent subcase. From this decomposition, one obtains a slice characterization, a criterion for separatedness, a comparison with ordinary holomorphic functions of two complex variables, active-variable Cauchy formulas and estimates, local series with parameter-dependent coefficients, reflection symmetry, and Hardy and Bergman kernel lifts on the separated Hilbert spaces.